Question

a doctor wanted to study the effect of four different treatments on mental health. a group of 100 adults experiencing depression volunteered for the study. the doctor randomly assigned one - fourth of them to each of four groups. group 1 followed a specific exercise plan, group 2 followed a specific diet plan, group 3 followed an exercise and diet plan, and group 4 did not follow any plan. after 4 weeks, the doctor contacted each participant and asked them if they felt any better. the results are displayed in the table.
the doctor would like to know if these data provide convincing evidence that the distribution of responses differs across the treatment groups in the population of all patients like these. are the conditions for inference met?
no, the random condition is not met.
no, the 10% condition is not met.
no, the large counts condition is not met.
yes, all three conditions for inference are met.

Explanation:

Step1: Check Random Condition

The doctor randomly assigned participants to groups. So random condition is met.

Step2: Check 10% Condition

We assume the population of all patients like these is much larger than 100 (since \(10\times100 = 1000\)). So 10% condition is met.

Step3: Check Large Counts Condition

Expected count formula: \(E=\frac{(\text{row total})(\text{column total})}{\text{grand total}}\).
Row totals: Yes - \(19 + 15+21 + 5=60\), No - \(6+10 + 4+20 = 40\). Column totals: Exercise Only - \(19+6 = 25\), Diet Only - \(15 + 10=25\), Both - \(21+4 = 25\), Neither - \(5+20=25\). Grand total \(n = 100\).
For each cell: \(E=\frac{60\times25}{100}=15\) (for yes - each treatment) and \(E=\frac{40\times25}{100} = 10\) (for no - each treatment). All expected counts (\(10\) and \(15\)) are \(\geq5\). So large counts condition is met.

Answer:

Yes, all three conditions for inference are met.